Input converter and sign discriminator for multipliers



April 1963 A. NATHAN 3,084,862

INPUT CONVERTER AND SIGN DISCRIMINATOR FOR MULTIPLIERS Filed Dec. 23,1958 2 Sheets-Sheet 1 FIG. 2

A ril 9, 1963 A. NATHAN 3,084,862

INPUT CONVERTER AND SIGN DISCRIMINATOR FOR MULTIPLIERS Filed Dec. 25,1958 2 Sheets-Sheet 2 FIG. 3

United States Patent 3,084,862 INPUT CQNVERTER AND SIGN DISCRRMINATGRFOR MULTIPLIERS Amos Nathan, 17 Lamed Heir Ave, Ramoth Rernez, Haifa,Israel Filed Dec. 23, 1958, Ser. No. 782,497 12 Claims. (Cl. 235194)This invention relates to an input converter and sign discriminator forelectronic multipliers. More specifically, this invention relates to adevice which, when connected to the input of any two-quadrant electronicmultiplier, or to the input of some one-quadrant multipliers, permitsfour-quadrant operation. This invention also provides a unit which, whenfed 'by two voltages, generates an output voltage whose sign is equal tothe sign of the product of the two input voltages.

In general, an electronic multiplier is a device, which when fed withinput voltages x and y, generates an output voltage which isproportional to xy. The following terminology is used to describe someaspects of the input ranges of multipliers: A multiplier is said tooperate in the first quadrant, if x20, ygtl; in the second quadrant, ifxO, 3'50; in the third quadrant, if x50, ytl; and in the fourth quadrantif x20, ygO.

-A multiplier is said to have one quadrant or two quadrant operation ifthe permissible input range is restricted to one quadrant or to twoadjacent quadrants, respectively.

The prior art provides input converters for oneor twoquadrantmultiplication whose operation is such, that the correct sign of theoutput of the multiplier is not always preserved. For example, in aconverter for a one quadrant multiplier operative in the first quadrant,x is converted into the modulus of x, ]x[, and y is converted into [y].In this case the sign of multiplier output is always non-negative.

It is an object of this invention to provide an input converter, which,when connected to the input terminals of any electronic multiplier,preserves both the value and the sign of the output of the multiplierand extends the operative range of one or both of the input variables inthe case of multipliers which do not have four quadrant operation.

. In particular, it is an object of this invention to provide twoquadrant input voltages to an electronic multiplier, when fed with fourquadrant inputs.

It is another object of this invention to provide means for producing anoutput voltage whose sign is equal to the sign of the product of twoinput voltages.

It is a further object of this invention to provide a device which, whenused with conventional additional circuits, converts any one quadrantmultiplier into a four quadrant multiplier.

It is yet another object of this invention to provide a device which,when used with additional conventional circuits, converts some halfquadrant multipliers into four quadrant multipliers.

It is another object of this invention to provide an input converter formultidimensional function generators with certain symmetry properties. A

yF-urther objects and advantages of this invention will become apparentfrom the following description of a practical realization of theinvention taken in connection with the accompanying drawings, in which-FIGURE 1A is a three dimensional plot of converter output 1 as afunction of its inputs x and y;

FIGURE 1B is a three dimensional plot of converter output .5 as afunction of its inputs x andy;

FIGURE 2 is a circuit diagram of one embodiment of the converter andsign discriminator of this invention; FIGURE/3 is a schematic diagram ofone embodiment 3,084,862 Patented Apr. 9, 1963 1= l l I I or,equivalently, as

using the sign dis- In FIGUR-E 1B the equation representing plane EOF is=x; the equation representing plane FOG is 5: the equation representingplane GOH is =x; and the equation representing plane HOE is .5: y.Surface EFGHO can therefore be represented as The terms maximum andminimum are used in this specification to denote selection operators;thus max. (a, b, c is equal to the greatest of a, b, c, and min. (a, b,c, is equal to the least of a, b, c,

' Any pair of non-zero values of x and y having differing moduli must ofnecessity satisfy one, and only one, of the following four conditions:

(a) y cg y x y y y y y Evaluating the functions 5 and 1 for each ofthese cases separately there is obtained respectively:

It follows, for all values of x and y,

(iv) The sign'of .5 is equal to the sign of xy A device generating g, 1as output signals when fed' with x, y as input signals can therefore beused as an input converter for a multiplier, 5 and 1 being the input,

signals to the multiplier; because from (i), above, the

output of the multiplier is not affected by the introductionof such aconverter in the said manner. Moreover, because of (ii), above, themultiplier need only be of the two quadrant type, whereas the converteraccepts inputs in four quadrants. The multiplier can actually be of aneven more restricted type, for (iii), above shows that input signals 5,7 lie within one quadrant only, said quadrant being limited by the lines23:1; and .=-a7. The part of the converter generating the .5 signal is asign discriminator for the sign of xy, as tollows from (iv), above.

FIGURE 2 is a schematic diagram of one embodiment of the device for thegeneration of 5 and 7;. T and T are the input terminals for inputsignals x and y, respectively. Input signal x is fed to sign changer Awhose output signal x appears at output terminal T Input signal y is fedto sign changer A whose output signal -y appears at output terminal TThe anodes of diodes D D D D are connected to terminals T T T Trespectively. Their cathodes are joined at terminal T which is alsoconnected to a negative D.C. voltage, at through resistor R Except for asmall transitional interval, only one of diodes D D D D conducts at atime; there thus appears at terminal T a signal approximately equal to 1If the diodes have a voltage drop equal to s when conducting, thevoltage at T will be equal to +6. When using silicon or germanium diodesor transistors, e is of the order of 0 .1 to 1.0 volt. To correct forthis offset voltage and for diode drift, diode D of the same type asdiodes D to D is connected with its cathode to T The anode of D isconnected to output terminal T and to a positive D.C. voltage, shown atthrough resistor R The offset voltage of D cancels the offset voltagesof diodes D to D and voltage 1; thus appears at T This compensationdevice, comprising D and R is needed only when high accuracy isrequired. Another way of compensating for said offset voltage consistsof replacing D by a resistor having a resistance which is equal to theresistance of D when conducting.

Some types of diodes have considerable spread of voltage drops fromdiode to diode, when conducting. The errors thus introduced into theconverter can be eliminated by first determining the voltage drop ofthat diode which has the largest drop, 6, when conducting, and thenadding resistors of suitable values in series with each of the otherdiodes so that the drop across each series combination of said resistorsand diodes is made equal to e when the diodes are conducting. Thiscompensation should be carried out for all diodes of the converter.

The cathodes of diodes D D D D are connected to terminals T T T Trespectively. The anodes of D and D are joined to T which is alsoconnected to a positive D.C. voltage, through a resistor R At T thusappears a signal equal to min.(x, y)+e. The anodes of diodes D and D arejoined at T, which is also connected to a. positive D.C. voltage,through resistor R At T; thus appears a signal equal to The anodes ofdiodes D D are connected to T T respectively. Their cathodes are joinedat terminal T which is also connected to a negative voltage, shown atthrough resistor R At '1; thus appears a signal equal to g. A twoquadrant multiplier, or a single quadrant multiplier whose inputs liewithin-the quadrant yglxl, can therefore be connected to terminals T andT,, No compensating diodes are required for the offset voltages of thediodes associated with the circuit for generating :3, because, by virtueof the configuration, the offset voltages cancel.

The device generating 5 heretofore described is based on the expressionAnalogously, a device with identical functions can be constructed, basedon the given equivalent form for .5

The description above is to be understood by way of example only.

FIGURE 3 shows one embodiment of the use of the sign discriminator ofthis invention to convert a one quadrant multiplier to four quadrantoperation. In FIGURE 3, T T T T are the terminals which are at voltagesx, x, y, y, respectively, as in the device described in connection withFIGURE 2. Terminal T; is at voltage 5. This voltage is generated by useof the part of the converter generating 5, as heretofore described, i.e.by use of the device shown in FIGURE 2 less that part of it consistingof D D D D R and T1,, and their interconnections. 111 FIGURE 3, M is aone quadrant multiplier whose input terminals are T and T T T are fedwith volta es ]x[, ]y[, respec tively. The anodes of diodes D D areconnected to T T respectively; their cathodes are connected to T T isalso connected to a negative D.C. voltage, through resistor R thevoltage of T is approximately equal to Diode D is provided in order tocompensate for the voltage drop in D and D and to compensate for diodedrift. The cathode of D is connected to T and its anode is connected toT which is also connected to a positive DC. voltage, through resistor RT is thus at a voltage equal to The anodes of diodes D D are connectedto T T respectively; their cathodes are connected to T which is alsoconnected to a negative D.C. voltage, through resistor R The cathode ofcompensating diode D is connected to T and its anode is connected toterminal T which is also connected to a negative D.C. voltage, throughresistor R T is therefore at a voltage equal to |y|. M is therefore fedwith non-negative voltages and can be of one quadrant operation. At theoutput terminal T of M thus appears a voltage equal to klxy], where k isa positive constant. Analogous considerations apply when k is negativeconstant. This voltage is fed to sign changer A whose output terminal Tis therefore at voltage kIxy]. Voltage 5 is fed from terminal T to apolarized relay whose actuating mechanism R connects terminal T of itschange-over switch S to T or T for positive or negative values of 5,respectively. At T therefore appears the required output voltage kxy. Cis a capacitor, connected between T and ground; C is required in someapplications, depending upon the circuit fed from T in order effectivelyto ground T during the switching over of S. The relay consisting of Rand S and their associated circuits is by way of illustration only; itcan be replaced, in particular, by an electronic switching device.

Similarly, a half quadrant multiplier can be converted into a fourquadrant multiplier. For example, to convertto four quadrant operation amultiplier operative in the half quadrant 3 2x20, said multiplier can befed with signals I5] and 1;. Referring again to FIGURE 3, this can bedone by omitting D D D R and R and their interconnections, and feeding TT T with E, 5, and 1 respectively, Where 5 and 1 are generated asdescribed above, whereas 5 can be obtained from 5 by sign changingmeans, or directly from x, x, y, -y from the expression The circuit forthe generation of 5 according to this later expression is thereforeidentical with the circuit of FIGURE 2 for the generation of 5, exceptthat anode and cathode connections of each of diodes D to D and thepositive and negative voltages at and respectively, must beinterchanged, and, because of the loading of the circuits generating aand 5, in the circuit generating 5 resistor R and the negative voltageto which it connects must be omitted, and in the circuit generating 5resistor R must be halved. E or E are also used as control signals foroutput selection.

Alternatively [til can be generated according to expressions such as forexample, .and the manner of said generations will be quite clear fromthe above description.

In one embodiment of the invention the following components andcomponent values were used:

All diodes are silicon junction diodes;

All voltages marked are 250 volts D.C.;

All voltages marked are 250 volts D.C.

In the device described in connection with FIGURE 2,

In the device described in connection with FIGURE 3,

R =R =2S0 kilohms R7=Rg=500 kilohms C 200' picofarads In someapplications, inputs x and/or 2 are restricted, or the multiplier hasspecial input characteristics. It is then not always required to realizethe complete surfaces of FIGURES 1A and 1B.

For example, if input variable y is always non-negative, 3720, theprevious expressions for 1; and 5 simplify to and the elements D D and Amay be omitted from the device of FIGURE 2.

Another example which realizes only a part of each of the surfaces ofFIGURE 1 is required if the multiplier is such that it can accept onlyinputs x and y for which ygx, the later restriction defining aparticular two quadrant multiplier, said two quadrants covering half ofthe first, the whole second, and half of the third quadrants.

One embodiment of the modification sufficient in the latter caseconsists of the unit of FIGURE 2 with the omission of the followingcomponents: A A D D D D D and R and their interconnections.

Other modifications, realizing other parts of the surfaces of FIGURE 1are obtained by the omission of selected components of the embodiment ofthis invention described in connection with FIGURE 2. 1

While the converter is described above as an input unit to a multiplier,it can also be used as an input unit to a two dimensional functiongenerator, or to any two inputs of a multidimensionalfunction'generator; in the latter case several converters can be used asinput units, each converter serving as input unit to one pair of inputterminals of the function generator or its other input con-' verters, orto a pair of terminals one of which is an input terminal of the functiongenerator and the other one is an input terminal of an input converter,all this, provided the function generated has the same symmetryproperties and/ or the same input restrictions as a multiplier.

For example, if f(x, y) is the function generated by the functiongenerator when it is fedwith voltages at and y, at its two inputterminals, and if fi y)=f(y )=f( y)=f(y the unmodified input convertercan be used and only one quadrant of f(x, y) need be realized in thefunction generator, said quadrant lying partly in the first and partlyin the second quadrants, between x=y and x: y.

As a second example, if (x, y) =f(y, x), the modification of thisinvention described above in connection with multipliers for which ygxcan be used as an input converter to the function generator and only twoquadrants of f(x, y) need be realized in the function generator, saidtwo quadrants lying partly in the first and third quadrants, andcovering the whole second quadrant, their limit being given by the linex=y.

As a third example, if

[-5 I and a can be used as input signals and output switching must beprovided, as described above in connection with the conversion of a halfquadrant multiplier to four quadrant operation.

The embodiments of this invention and its modifications which have beendescribed in connection with FIGURE 2 use diodes as non-linear elements.It is to be understood that the use of diodes is by way of example only;in particular, the diodes can be replaced by transistors connected in aso-called diode logic connection. The manner of said replacement will bedescribed in connection with FIGURES 4A to 4D, in which FIGURE 4Arepresents the diagram of a typical diode, such as D D D D D or D ofFIGURE 2, together with resistor R connected to its cathode, whichcorresponds to R or R in the case of D D D D or D D respectively. 3 isthe terminal of the constant negative voltage, FIGURE 4B shows thecorresponding transistor connection, in which terminals 1, 2, 3 areidentical with terminals 1, 2, 3, of FIGURE 4A. TR is an NPN transistor.The parallel connection of resistor R and capacitor 6 serves as inputnetwork to the base of the transistor. The base is also connected tobias voltage V at terminal 4 through resistor R The emitter is connectedto terminal 2 which is also connected to a negative D.C. voltage V atterminal 3, through resistor R The collector is connected to terminal 5which is at the constant voltage V FIGURE 40 corresponds to one of thediodes such as D D D D D in the embodiment of FIGURE 2, and consists ofFIGURE 4A with diode 5, replaced by diode 1 5 where D is D withinterchanged terminals, and with voltage V of reversed sign. FIGURE 4Dshows the transistor equivalent to FIGURE 4C and is identical to FIGURE4B except for the substitution of PNP transistor TR for TR and thereversal of sign of voltages V V and V Combinations of transistors anddiode elements can be used. For example, D 10, D may be diodes, allothernon-linear elements being transistors.

, The use of transistors has the main advantages of providing impedanceconversion, from a higher input to a lower output value, and ofimproving the frequency response; its main disadvantage is therestriction of, the useful voltage range of input voltages at and y.

, Typical values for the embodiments of FIGURES 4B and 4 D are asfollows: Rf =50 kilohms; R =2.0 kilohms; E =10O kilohms; 0:570picofarads. Typical voltages in the embodiment of FIGURE 48 are asfollows; V =22.5 volts; V =7 volts; V =2 volts. In the embodiment ofFIGURE 4D the same voltage values with reversed sign are used. The gainfrom input to output was found to be 0.98. The permissible voltage rangeat terminal Zwas 3 volts to +3 volts. Adjustment for zero output voltagewith zero input voltage is obtained by a slight variation of V For thispurpose adjustable V is provided.

As an alternative way of carrying out the invention, 5 as given aboveand 1;=min.(x, -x, y, -y) may be used, or, for example, realization ofand 7 according to the expressions, which require only one sign-changer,

andthe manner of their generation will be quite clear from the abovedescription.

.What I claim is:

1. An input converter for a one or two quadrant multiplier or likefunction generator, comprising means for accepting first and secondinput signals, first means connected to said accepting means forproducing from said input signals a first output signal correspondingwith the maximum of the moduli of said input signals, and second meansconnected to said accepting means for producing a second output signalhaving a modulus equal to the minimum of the moduli of the inputsignals, said second means including means for generating a sign forsaid second output signal wherein said sign is equal to the sign of theproduct of the input signals; said accepting means including means forgenerating a plurality of secondary output signals greater in numberthan said input signals; at least one of said secondary output signalsbeing equal in mag nitude and opposite in sign to one of said inputsignals; said first and second means being adapted to employ saidsecondary output signals to generate appropriate moduli signals.

2. The input converter of claim 1 wherein the means for producing thefirst output signal comprises first and second sign changer means forproducing first and second secondary input signals equal to the inversesof the first and second input signals, and third means for selecting themaximum of said first and second input signals and said first and secondsecondary input signals.

'3. An input converter for a one or two quadrant multiplier or likefunction generator, comprising means for accepting first and secondinput signals, first and second sign changer means for producing firstand second secondary input signals equal to the inverse of therespective input signals, means for selecting the maximum of said firstand second input signals and said first and second secondary inputsignals, means for selecting the minimum of said first and second inputsignals, means for selecting the minimum of said first and secondsecondary input signals, and means for selecting the maximum of the twominima so selected.

4. A four quadrant multiplier comprising means for accepting first andsecond input signals, means for deriving the moduli of said inputsignals, a single quadrant multiplier for multiplying said moduli toproduce a first output signal, sign changer means to produce a secondoutput signal equal to the inverse of said first output signal, andchangeover means for selecting one of said first and second outputsignals as final output signal, said changeover means being responsiveto the sign of a control signal derived from said input signals so thatthe output signal selected has the sign as well as the modulus of theproduct of the input signals.

5. A four quadrant multiplier comprising means for accepting first andsecond input signals, means for deriving the moduli of said inputsignals, means for multiplying said moduli to produce a first outputsignal, means for changing the sign of said first output to produce asecond output signal, means for deriving a control signal having thesign of the product of the input signals, and changeover means,responsive to said control signal, for selecting that one of said firstand second output signals having the sign of said control signal.

6. A four quadrant multiplier or like function generator comprisingmeans for accepting first and second input signals, means for deriving afirst secondary signal equal to the maximum of the moduliof the inputsignals, means for deriving a second secondary signal equal to theminimum of the moduli of the input signals, a half quadrant multiplieror like function generator .producing a first output signal from saidsecondary signals, sign changer means to produce a second output signalequal to the inverse of said first output signal, and changeover meansfor selecting one of said first and second output signals as finaloutput signal, said changeover means being responsive to the sign of acontrol signal derived from said input signals so that the output signalselected has the sign of the product of the input signals.

7. A multiplier or function generator for the generation of the modulusof a product or of another function, comprising means for acceptingfirst and second input signals, means for deriving a first secondarysignal equal to the maximum of the moduli of the input signals, meansfor deriving a second setiondary signal equal to the mini- 8 mum of themoduli of the input signals, a halt quadrant multiplier or like functiongenerator producing an output signal from said secondary signals, saidoutput signal being the required modulus.

8. A multiplier as set forth in claim 5 wherein the means for derivingsaid control signal comprises first and second sign changer means forproducing first and second secondary input signals equal to the inversesof the first and second input signals respectively, means for selectingthe minimum of said first and second input signals, means for selectingtheminimum of said first and second secondary input signals and meansfor selecting the maximum of the minima so selected, said maximum beingthe required control signal.

9. In a multiplier or like function generator, means for deriving asignal with the same sign as the product of two input signals,comprising first and second sign changer means for producing first andsecond secondary input signals equal to the inverses' of the first andsecond input signals respectively, means for selecting the minimum ofsaid first and second input signals, means for selecting the minimum ofsaid first and second secondary input signals and means for selectingthe maximum of the minima so selected, said maximum being the requiredsignal.

'10. A sign discriminator for use in multipliers or similarlysymmetrical function generators, comprising means for accepting firstand second input signals, first and second sign changer means forproducing first and second secondary input signals equal to the inversesof the first and second input signals respectively, means for selectingthe minimum of said first and second input signals, means for selectingthe minimum of said first and second secondary input signals, and meansfor selecting the maximum of the minima so selected, said maximum havingthe sign of the product of the two input signals.

11. An input converter for a one or two quadrant multiplier or likefunction generator, comprising means for accepting first and secondinput signals, means for producing from said input signals a firstoutput signal correspond-ing with the maximum of the moduli of saidinput signals, and means for producing a second output signal having amodulus equal to the minimum of the moduli of the input signals and asign equal to the sign of the product of the input signals, the meansfor producing the second output signal comprising first and second signchanger means for producing first and second secondary input signalsequal to the inverses of the first and second' input signalsrespectively, means for selecting the minimum of said first and secondinput signals, means for selecting the minimum of said first and secondsecondary input signals and means for selecting the maximum of theminima so selected, said maximumbeing the required second output signal.

12. An input converter for a one or two quadrant multiplier or likefunction generator, comprising means for accepting first and secondinput signals, means for producing from said input signals a firstoutput signal corresponding with the maximum of the moduli of said inputsignals, and means for producing a second output signal having a modulusequal to the minimum of the moduli of the input signals and a'sign equalto the sign of the product of the input signals, the means for producingthe second output signal comprising first and second sign changer meansfor producing first and second secondary input signals equal to theinverse of the first and second input signals respectively, means forselecting the maximum of said first input signal and said secondaryinput signal, means for selecting the maximum of said second inputsignal and said first secondary input signal and means for selecting theminimum of the maxima so selected, said minimum being the requiredsecond output signal.

(References on following page) References Cited in the file of thispatent UNITED STATES PATENTS Van Allen Oct. 8, 1957 OTHER REFERENCESElectronic Analog Computer, GAP/R K2 Series, Philbrick Researches, Inc.,received in Scientific Library 10 Sept. 29, 1958. FIG. 1.1, p. 10 andFIG. 4.3, p. 23 relied on.

Electronic Analog Computer (Korn and K0111), 1952, McGraw-Hill BookCompany, Inc, N.Y., FIG. 6.2, page 212 showing adaptation oftwo-quadrant multipliers to four-quadrant multiplication of interest.

1. AN INPUT CONVERTER FOR A ONE OR TWO QUADRANT MULTIPLIER OR LIKEFUNCTION GENERATOR, COMPRISING MEANS FOR ACCEPTING FIRST AND SECONDINPUT SIGNALS, FIRST MEANS CONNECTED TO SAID ACCEPTING MEANS FORPRODUCING FROM SAID INPUT SIGNALS A FIRST OUTPUT SIGNAL CORRESPONDINGWITH THE MAXIMUM OF THE MODULI OF SAID INPUT SIGNALS, AND SECOND MEANSCONNECTED TO SAID ACCEPTING MEANS FOR PRODUCING A SECOND OUTPUT SIGNALHAVING A MODULUS EQUAL TO THE MINIMUM OF THE MODULI OF THE INPUTSIGNALS, SAID SECOND MEANS INCLUDING MEANS FOR GENERATING A SIGN FORSAID SECOND OUTPUT SIGNAL WHEREIN SAID SIGN IS EQUAL TO THE SIGN OF THEPRODUCT OF THE INPUT SIGNALS; SAID ACCEPTING MEANS INCLUDING MEANS FORGENERATING A PLURALITY OF SECONDARY OUTPUT SIGNALS GREATER IN NUMBERTHAN SAID INPUT SIGNALS; AT LEAST ONE OF SAID SECONDARY OUTPUT SIGNALSBEING EQUAL IN MAGNITUDE AND OPPOSITE IN SIGN TO ONE OF SAID INPUTSIGNALS; SAID FIRST AND SECOND MEANS BEING ADAPTED TO EMPLOY SAIDSECONDARY OUTPUT SIGNALS TO GENERATE APPROPRIATE MODULI SIGNALS.